For the isentropic compressible Euler equations (REF ),
Feireisl-Gwiazda-Gwiazda-Wiedemann [1]} show that
\(\begin{aligned}&\quad 0 \le \underline{\varrho } \le \varrho \le \bar{\varrho } \text{ a.e. in }(0, T) \times \mathbb {T}^{d},\pi \in C^{2}[\varrho , \bar{\varrho }]\\&\varrho , \varrho v \in B_{3, \infty }^{\beta } ((0, T) \times \mathbb {T}^{d} ), v \in B_{3, \infty }^{\alpha } ((0, T) \times \mathbb {T}^{d} ),\end{aligned}\) 
